verification: implicit contact patch test and Hertz line contact

Two oracle cases: the contact patch test on non-matching meshes in small and finite sliding,
and Hertz's cylinder on a flat under force control (peak pressure, pressure distribution,
subsurface stress). docs/fem.md describes the formulation, the literature survey its papers;
ROADMAP M7 and parity are updated.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01UhhZMBTviDQxZvwnyP89fo

docs/oracle.md

@@ -2,7 +2,7 @@
 
 Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the policy is [verification.md](verification.md). Every case is our own model, built with FieldCraft commands from the problem's published description, and compared with the reference value its `expected.json` cites, within the stated tolerance and for the stated reason. Accepted results are frozen in each case's `fixture.json`: a later change that moves one beyond its band (10⁻⁹ of its size) fails CI even while the reference still passes. No reference value was produced with a commercial product.
 
-**201 of 207 quantities pass** (6 known failure(s)) in 62 active cases: 53 theory, 9 published benchmark, 0 experiment; 10 pending cases wait for capabilities still being built.
+**222 of 228 quantities pass** (6 known failure(s)) in 64 active cases: 55 theory, 9 published benchmark, 0 experiment; 10 pending cases wait for capabilities still being built.
 
 ## Theory: closed-form solutions, patch tests and mesh invariants
 
@@ -80,6 +80,21 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
 | [lame-cylinder-plane-strain](../tests/oracle/lame-cylinder-plane-strain/) | radial displacement at the bore (10, 0) | 0.0090794 | 0.0090793 | -0.000267% | 0.1% | pass | matches |
 | [lame-cylinder-plane-strain](../tests/oracle/lame-cylinder-plane-strain/) | hoop stress σ_θ = σ_yy at the bore (10, 0) | 166.67 | 166.36 | -0.183% | 1% | pass | matches |
 | [lame-cylinder-plane-strain](../tests/oracle/lame-cylinder-plane-strain/) | axial stress σ_zz = 2νA at the bore (plane strain) | 20 | 20.083 | 0.415% | 1% | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_zz of the lower block's corner element under the interface (element 10) over that of the upper block's corner element over it (element 19) | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_zz of the lower block's centre element under the interface (element 14) over that of element 19 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_zz of the lower block's bottom corner element (element 1) over that of element 19 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_zz of the upper block's top far corner element (element 26) over that of element 19 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_xx of element 14 over its σ_zz (uniaxial stress: 0) | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-6 | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: the contact pressure at the secondary node (0.5, 0.5, 1) over −σ_zz of element 19 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: the integrated contact pressure over the reaction at the pushed top | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: σ_zz of the lower block's corner element under the interface (element 10) over that of the upper block's corner element over it (element 19) | 1 | 1 | 0.000151% | 0.02% | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: σ_zz of the lower block's centre element under the interface (element 14) over that of element 19 | 1 | 0.99999 | -0.00129% | 0.02% | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: σ_zz of the lower block's bottom corner element (element 1) over that of element 19 | 1 | 0.99999 | -0.000942% | 0.02% | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: σ_zz of the upper block's top far corner element (element 26) over that of element 19 | 1 | 0.99999 | -0.000872% | 0.02% | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: σ_xx of element 14 over its σ_zz (uniaxial stress: 0) | 0 | ≈ 0 | ≈ 0 | ± 2.0000e-4 | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: the contact pressure at the secondary node (0.5, 0.5, 1) over −σ_zz of element 19 | 1 | 0.99998 | -0.00172% | 0.02% | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: the integrated contact pressure over the reaction at the pushed top | 1 | 1 | < 1e-7% | 0.02% | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_zz of element 19 (uniaxial stress E ε = −10 for the 1% squeeze) | -10 | -9.9996 | 0.00426% | 0.01% | pass | matches |
 | [nl-cylinder-inflation](../tests/oracle/nl-cylinder-inflation/) | radial displacement of the bore (10, 0, 0) at the pressure that inflates it to a = 15 | 5 | 5.004 | 0.0793% | 0.5% | pass | matches |
 | [nl-cylinder-inflation](../tests/oracle/nl-cylinder-inflation/) | radial displacement of the outer wall (20, 0, 0) | 2.9129 | 2.9173 | 0.152% | 0.5% | pass | matches |
 | [nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/) | tip deflection δ_v/L at P L²/(E I) = 1 | 0.30172 | 0.30226 | 0.179% | 1% | pass | matches |
@@ -90,6 +105,12 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
 | [nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/) | tip shortening δ_h/L at P L²/(E I) = 5 | 0.38763 | 0.38955 | 0.495% | 2% | pass | matches |
 | [nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/) | tip deflection δ_v/L at P L²/(E I) = 10 | 0.81061 | 0.81421 | 0.444% | 1% | pass | matches |
 | [nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/) | tip shortening δ_h/L at P L²/(E I) = 10 | 0.555 | 0.558 | 0.541% | 2% | pass | matches |
+| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the largest contact pressure p₀ (at x = 0) | 2884.6 | 2892 | 0.254% | 2% | pass | matches |
+| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the force across the contact (half model, 1 mm thick) | 1132.8 | 1133 | 0.0221% | 0.2% | pass | matches |
+| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the contact pressure at the secondary node at x = 0.29594 (node 7) | 2325.1 | 2257.8 | -2.89% | 5% | pass | matches |
+| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the contact pressure at the secondary node at x = 0.36701 (node 8) | 1959 | 2078.4 | 6.09% | 10% | pass | matches |
+| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the contact pressure at the secondary node at x = 0.63325 (node 11, beyond the half-width b = 0.5) | 0 | ≈ 0 | ≈ 0 | ± 1 | pass | matches |
+| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the vertical stress σ_yy under the centre of contact at depth 0.47127 (node 713) | -2099.2 | -2059.8 | 1.87% | 4% | pass | matches |
 | [nl-hybrid-incompressible](../tests/oracle/nl-hybrid-incompressible/) | axial Cauchy stress, stretch λ = 2, ν = 0.4999 | 3.4988 | 3.4988 | 0.000000135% | 1e-6 rel. | pass | matches |
 | [nl-hybrid-incompressible](../tests/oracle/nl-hybrid-incompressible/) | lateral displacement at (1, 1, 1): l − 1 | -0.29281 | -0.29281 | < 1e-7% | 1e-6 rel. | pass | matches |
 | [nl-j2-uniaxial](../tests/oracle/nl-j2-uniaxial/) | axial Cauchy stress at a logarithmic strain of 0.2 | 642.74 | 642.74 | < 1e-7% | 1e-6 rel. | pass | matches |
@@ -259,8 +280,10 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
 - **[nafems-le1](../tests/oracle/nafems-le1/)** (benchmark): NAFEMS LE1 elliptic membrane (hex20 plane stress layer). Source: NAFEMS, The Standard NAFEMS Benchmarks, TNSB Rev. 3, October 1990, test LE1 (target σ_yy = 92.7 MPa at D); as reproduced in the Abaqus 2017 Benchmarks Guide, 'LE1: Plane stress elements—elliptic membrane'. Tolerances — syy_D: the target is published to three figures; plane stress modelled by a thin solid layer and the stress extrapolated to the edge add a few tenths of a percent
 - **[nafems-le1-plane-stress](../tests/oracle/nafems-le1-plane-stress/)** (benchmark): NAFEMS LE1 elliptic membrane with 2D plane stress elements (quad8). Source: NAFEMS, The Standard NAFEMS Benchmarks, TNSB Rev. 3, October 1990, test LE1 (target σ_yy = 92.7 MPa at D); as reproduced in the Abaqus 2017 Benchmarks Guide, 'LE1: Plane stress elements—elliptic membrane'. Tolerances — syy_D: the target is published to three figures; the stress extrapolated to the edge on a converged mesh adds a few tenths of a percent
 - **[nafems-le10](../tests/oracle/nafems-le10/)** (benchmark): NAFEMS LE10 thick plate under pressure (hex20). Source: NAFEMS, The Standard NAFEMS Benchmarks, TNSB Rev. 3, October 1990, test LE10 (target σ_yy = −5.38 MPa at D); as reproduced in the Abaqus 2017 Benchmarks Guide, 'LE10: Thick plate under pressure'. Tolerances — syy_D: the target is published to three figures; at D the stress is extrapolated to a corner where the hole edge meets the loaded face, and published converged solutions scatter by 1–2% (Abaqus Benchmarks Guide fine meshes: −5.51 to −5.89)
+- **[nl-contact-patch](../tests/oracle/nl-contact-patch/)** (theory): Contact patch test: two blocks with non-matching meshes in uniaxial compression (implicit mortar contact). Source: The contact patch test: a contact discretisation must transmit a uniform pressure across a non-matching interface exactly; Taylor & Papadopoulos, On a patch test for contact problems in two dimensions, Nonlinear Computational Mechanics, Springer 1991, 690–702; Crisfield, Re-visiting the contact patch test, IJNME 48:435–449, 2000; passed by mortar methods: Puso & Laursen, CMAME 193:601–629, 2004. Uniaxial stress between frictionless symmetry planes is uniform in both blocks. Tolerances — small_lower_corner: small sliding is linear and its mortar integrals exact on these rectangular faces: a uniform stress is the discrete solution up to the convergence tolerance (10⁻¹⁰); small_lower_centre: as for the corner element; small_lower_base: as for the corner element; small_upper_top: as for the corner element; small_lateral: frictionless symmetry planes and a frictionless interface leave the blocks free to expand sideways; the non-matching interface must not hold them; small_pressure: the mortar multipliers are the nodal values of the contact pressure, uniform and equal to the stress across the interface; small_force: the pressure carries the whole load across the interface; finite_lower_corner: finite sliding is exact up to the geometric terms of the gaps the augmentation leaves (10⁻⁵ of the element thickness) and the default convergence tolerances; finite_lower_centre: as for the corner element; finite_lower_base: as for the corner element; finite_upper_top: as for the corner element; finite_lateral: frictionless symmetry planes and a frictionless interface leave the blocks free to expand sideways; the non-matching interface must not hold them; finite_pressure: the mortar multipliers are the nodal values of the contact pressure, uniform and equal to the stress across the interface; finite_force: the pressure carries the whole load across the interface; small_stress: the squeeze less the penetration the augmented Lagrangian leaves (up to 10⁻⁵ of the element thickness, a few 10⁻⁵ of the squeeze)
 - **[nl-cylinder-inflation](../tests/oracle/nl-cylinder-inflation/)** (theory): Inflation of a thick neo-Hookean cylinder by a follower pressure (plane strain). Source: Finite inflation of an incompressible neo-Hookean thick-walled tube in plane strain: r² = R² + a² − A², σ_θ − σ_r = μ (λ² − λ⁻²) with λ = r/R, integrated through the wall: p = (μ/2)[(ln λ_a² − λ_a⁻²) − (ln λ_b² − λ_b⁻²)] (R. S. Rivlin, Large elastic deformations of isotropic materials VI, Phil. Trans. R. Soc. A 242:173–195, 1949; R. W. Ogden, Non-Linear Elastic Deformations, Dover 1997, §5.3.2). Tolerances — ua: the closed form is incompressible and our material nearly so (κ/μ = 1000, a volume change of order p/κ ≈ 4·10⁻⁴); hybrid hex20, 6 × 16, follow the wall's strain gradient well inside; ub: as at the bore
 - **[nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/)** (theory): Large-deflection cantilever under a dead tip load against the elastica (solids). Source: The elastica of a cantilever under a vertical end load: K. E. Bisshopp, D. C. Drucker, Large deflection of cantilever beams, Quarterly of Applied Mathematics 3(3):272–275, 1945 (elliptic-integral solution); values computed by integrating θ″ = −(P/EI) cos θ by shooting (our own RK4, 20000 steps), agreeing with the table of K. Mattiasson, Numerical results from large deflection beam and frame problems analysed by means of elliptic integrals, IJNME 17(1):145–153, 1981, to all six digits. Tolerances — v1: hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica; u1: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's; v2: hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica; u2: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's; v5: hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica; u5: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's; v10: hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica; u10: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's
+- **[nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/)** (theory): Hertz line contact: a cylinder pressed on a flat (implicit mortar contact, plane strain). Source: H. Hertz, Über die Berührung fester elastischer Körper, J. reine angew. Math. 92:156–171, 1882; K. L. Johnson, Contact Mechanics, Cambridge University Press 1985, §4.3 (two-dimensional contact of cylindrical bodies, eqs. 4.45–4.49): b = √(4PR/(πE*)), p₀ = 2P/(πb), with 1/E* = 2(1 − ν²)/E for two bodies of one material and R the cylinder's radius on a flat. Tolerances — peak_pressure: p₀ = 2P/(πb) for the load per length P = 2265.57 N/mm, b = 0.5 mm; the contact width spans seven elements of the graded mesh; normal_force: the pressure carries the whole load; the soft springs that hold the indenter before contact take a millionth; pressure_x0296: p₀ √(1 − x²/b²) at the node; the nodal pressure (the node's normal force over the area it carries) averages the distribution over the node's elements; pressure_x0367: as at node 7, nearer the edge, where the elements round off the square-root fall of the pressure to zero at b; pressure_x0633: outside the contact the surfaces are apart: no pressure (an element beyond b, since the edge element carries part of the last closed node's pressure); syy_depth: σ_z = −p₀ / √(1 + z²/b²) on the axis of a half-plane (Johnson 1985, eq. 4.49); the foundation is ten half-widths wide and deep, not a half-plane, and the stress is the nodal average of the elements around the node
 - **[nl-hybrid-incompressible](../tests/oracle/nl-hybrid-incompressible/)** (theory): Nearly incompressible hybrid block under prescribed stretch. Source: Uniaxial stress of the decoupled compressible neo-Hookean energy W = μ/2 (Ī₁ − 3) + κ/2 (J − 1)² (P. J. Flory, Trans. Faraday Soc. 57:829–838, 1961): σ = μ J^(−5/3) (λ² − I₁/3) + κ (J − 1) with the lateral stretch found so σ_yy = 0 (bisection; E = 3, ν = 0.4999: μ = 1.0000667, κ = 5000). Tolerances — sigma: a homogeneous deformation every hybrid hex8 represents exactly; the 27-element mesh makes the global mixed tangent indefinite in the first iterations (LDLᵀ): the tolerance admits the Newton tolerance and rounding; lateral: a homogeneous deformation every hybrid hex8 represents exactly; the 27-element mesh makes the global mixed tangent indefinite in the first iterations (LDLᵀ): the tolerance admits the Newton tolerance and rounding
 - **[nl-j2-uniaxial](../tests/oracle/nl-j2-uniaxial/)** (theory): Uniaxial tension of a J2 bilinear-hardening cube to large strain (implicit, finite strain). Source: J2 plasticity with linear isotropic hardening in uniaxial stress (Simo and Hughes, Computational Inelasticity, Springer 1998, §2.3) in the logarithmic strain, which Simo's exponential return map (CMAME 99:61–112, 1992) integrates exactly on a proportional path: Kirchhoff stress τ = σ_y + E_t (ε − σ_y/E), Cauchy stress σ = τ/J with ln J = τ/(3K) (Hencky's volume law). Tolerances — sigma: a homogeneous deformation the element represents exactly, reached on a proportional path: the tolerance only admits the Newton tolerance and rounding; plastic_strain: ε_p = (τ − σ_y)/H, as for the stress
 - **[nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/)** (theory): Mooney–Rivlin block in uniaxial, equibiaxial and pure-shear tension. Source: Treloar's homogeneous modes of rubber testing (L. R. G. Treloar, The Physics of Rubber Elasticity, 3rd ed., Oxford 1975, ch. 10) for the decoupled compressible energy W = C₁₀ (Ī₁ − 3) + C₀₁ (Ī₂ − 3) + κ/2 (J − 1)²: principal Cauchy stresses σᵢ = (λᵢ/J) ∂W/∂λᵢ with the free stretch found so the free faces carry no stress (bisection; E = 3, ν = 0.49, C₀₁ = 0.2, C₁₀ = μ/2 − C₀₁). Tolerances — uniaxial: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; uniaxial_lateral: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; equibiaxial: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; shear_xx: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; shear_yy: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding